Challenges of Learning a Latent Variable Factorization of Pre-trained Language Models

This note records a writing-level framing of why the project is not just a standard conditional VAE setup. It sits between the paper-writing notes and the methodology/theory notes.

Related notes:

Relation to latent variable generative modeling

Our approach is related to latent variable generative modeling approaches such as VAEs. In a VAE, we observe data $x \sim D$ from some data distribution and learn a generator $p(x \mid z)$ that maps a latent variable $z$ to an instance. Conditional VAEs apply this to learning a conditional distribution $p(y \mid x)$ from a dataset of pairs $(x, y) \sim \mathcal{D}$ .

Central difference

Rather than learning from a dataset that defines the target conditional law, we are factorizing and adapting a pre-trained model that already models $p(y \mid x)$ . The problem is not just latent-variable modeling. It is transforming an already-capable model into a latent-factorized model whose latent variable corresponds to abstract strategies.

Core challenges

Learning a latent variable generative model over discrete sequence space with an autoregressive model class.
Working in the adaptation setting rather than the ordinary dataset-learning setting.

The second point is the more fundamental difference. Rather than learning from data with some dataset

\mathcal{D} = \{(x_i, y_i)\},

we start from a model that already captures the observable distribution $p(y \mid x)$ and ask for a useful latent factorization of that law.

Challenges introduced by this setting

The generative model already captures $p(y \mid x)$ and therefore does not need a latent variable in order to fit the observable law.
The distribution may already be a mixed and entangled superposition of multiple strategies, with the latent "modes" entangled in the hidden states.
Posterior collapse remains a major risk, especially because the decoder is strong enough to model much of the conditional law directly.
Adaptation data comes from the model rather than from an external dataset, which makes the setup overlap with distillation-like training even if "distillation" is not quite the right term.

Why this note matters

This framing explains why the project has to combine:

methodology for preventing trivial or collapsed latent usage,
theory for understanding non-identifiability and selection among many factorizations,
experiments that test whether recovered latent variables align with intended strategies rather than arbitrary predictive partitions.